Statistical Variability in Nonlinear Spaces: Application to Shape Analysis and DT-MRI

P. THOMAS FLETCHER: Statistical Variability in Nonlinear Spaces: Application to Shape Analysis and DT-MRI. (Under the direction of Stephen M. Pizer and Sarang Joshi.) Statistical descriptions of anatomical geometry play an important role in many medical image analysis applications. For instance, geometry statistics are useful in understanding the structural changes in anatomy that are caused by growth and disease. Classical statistical techniques can be applied to study geometric data that are elements of a linear space. However, the geometric entities relevant to medical image analysis are often elements of a nonlinear manifold, in which case linear multivariate statistics are not applicable. This dissertation presents a new technique called principal geodesic analysis for describing the variability of data in nonlinear spaces. Principal geodesic analysis is a generalization of a classical technique in linear statistics called principal component analysis, which is a method for computing an efficient parameterization of the variability of linear data. A key feature of principal geodesic analysis is that it is based solely on intrinsic properties, such as the notion of distance, of the underlying data space. The principal geodesic analysis framework is applied to two driving problems in this dissertation: (1) statistical shape analysis using medial representations of geometry, which is applied within an image segmentation framework via posterior optimization of deformable medial models, and (2) statistical analysis of diffusion tensor data intended as a tool for studying white matter fiber connection structures within the brain imaged by magnetic resonance diffusion tensor imaging. It is shown that both medial representations and diffusion tensor data are best parameterized as Riemannian symmetric spaces, which are a class of nonlinear manifolds that are particularly well-suited for principal geodesic analysis. While the applications presented in this dissertation are in the field of medical image analysis, the methods and theory should be widely applicable to many scientific fields, including robotics, computer vision, and molecular biology.